Very Computer

ANDs and ORs

DIGEST OF BUCKLEY, JJ AND SILER, W: A NEW T-NORM. SUBMITTED TO FUZZY SETS AND SYSTEMS, 1996.

Fuzzy systems theory has been criticized for not obeying all the laws of classical set theory and classical logic. The t-norm and t-conorm here presented obey all the laws of the corresponding classical theory. A somewhat similar theory has been proposed by Thomas (1994), except that he does not claim that the distributive property is maintained.

We first propose a source of fuzziness. We suppose that the truth value > 0 and < 1 of a fuzzy logical statement A is drawn from a number of underlying (probably implicit) correlated random variables from a Bernoulli process whose values alpha[i] are binary, i.e. 0 or 1 with a Bernoulli distribution, and that the truth value of A is a simple average of these binary values. (George Klir (1994) proposed a similar process where the random values are binary opinions of experts as to truth or falsehood of a statement.) If this is so, then

We now suppose that this basic process is inaccessible to us, but that we do have a history of a number of instances of the truths of statement A and statement B. Now, given a value of r, the correlation coefficient between a, the truth values of A, and b, the truth values of B, the t-norm and t-conorm appropriate to this history, T (t-norm) and C (t-conorm) are defined for [a, b] on S, a restricted subset of [0,1]x[0,1].

Theorem 4. (The 5 parts of this theorem define the subset S of [0,1]x[0,1] possible for r = 1, 0 < r < 1, r = 0, -1 < r < 0 and r = -1.) Given a value of r, it may be that not all (a,b) combinations are possible; e.g. a = .25, b = .75 is not possible for r = 1 in the binary process described above.)

Theorem 7. 1. A AND A = A, r is 1. 2. A OR 0 = 0, any r. 3. A OR X = A, any r. 4. A AND NOT-A = 0, r is -1. 5. A OR A = A, r is 1. 6. A OR X = A, any r. 7. A OR 0 = A, any r. 8. A OR NOT-A = X, r is -1. 9. NOT-(A AND B) = NOT-A OR NOT-B, any r. 10. NOT-(A AND B) = NOT-A AND NOT-B, any r. 11. A OR (A AND B) = A, any appropriate r. 12. A AND (A OR B) = A, any appropriate r. 13. A AND (B OR C) = (A AND B) OR (A AND C), any appropriate r. 14. A OR (B AND C) = (A OR B) AND (A OR C), any appropriate r.